Question

for each equation, determine whether it is linear.
equation | is the equation linear?
| yes | no
(a) $y = -x$ | $\circ$ | $\circ$
(b) $y = -4x^2 + 3$ | $\circ$ | $\circ$
(c) $y = 1$ | $\circ$ | $\circ$
(d) $y = 7x + 2$ | $\circ$ | $\circ$

Explanation:

Step1: Recall linear equation form

A linear equation in two variables is of the form \( y = mx + b \), where \( m \) and \( b \) are constants, and the highest power of \( x \) is 1.

Step2: Analyze equation (a) \( y = -x \)

Rewrite as \( y = -1x + 0 \). Here, \( m = -1 \), \( b = 0 \), and \( x \) has power 1. So it is linear. Select "Yes".

Step3: Analyze equation (b) \( y = -4x^2 + 3 \)

The power of \( x \) is 2, which is not 1. So it is not linear. Select "No".

Step4: Analyze equation (c) \( y = 1 \)

Rewrite as \( y = 0x + 1 \). Here, \( m = 0 \), \( b = 1 \), and \( x \) has power 1 (implicitly, \( x^1 \) with coefficient 0). So it is linear. Select "Yes".

Step5: Analyze equation (d) \( y = 7x + 2 \)

This is in the form \( y = mx + b \) with \( m = 7 \), \( b = 2 \), and \( x \) has power 1. So it is linear. Select "Yes".

Answer:

(a) \( y = -x \): Yes
(b) \( y = -4x^2 + 3 \): No
(c) \( y = 1 \): Yes
(d) \( y = 7x + 2 \): Yes